The Role Of The Sampling Distribution In Understanding .
Mathematics Education Research Journal2003, Vol. 15, No. 3, 270-287The Role of the Sampling Distribution inUnderstanding Statistical InferenceKay LipsonSwinburne University of TechnologyMany statistics educators believe that few students develop the level of conceptualunderstanding essential for them to apply correctly the statistical techniques at theirdisposal and to interpret their outcomes appropriately. It is also commonly believedthat the sampling distribution plays an important role in developing thisunderstanding. This study clarifies the role of the sampling distribution in studentunderstanding of statistical inference, and makes recommendations concerning thecontent and conduct of teaching and learning strategies in this area.Over recent years there has been an expansion in the teaching of statistics at alleducational levels. This increase has been due, in part, to the recognition of theimportance of quantitative literacy but also to the availability of computer-basedtechnology that can be used to carry out sophisticated statistical analyses. Manyfeel, however, that although increasing numbers of students study statistics, thenumber who gain a real appreciation of the power and purpose of statistics isextremely small (see, e.g., Garfield, 2002; Konold, 1991; Williams, 1998). To datethere is little empirical evidence that increasingly refined technological support isdoing much to change this (Mills, 2002). More research is needed in order toimprove the structure and teaching of introductory statistics.The focus of this research was the statistical concepts that are critical to anunderstanding of statistical inference, in particular the teaching and learning of thesampling distribution. Clearly, this concept is of fundamental statisticalimportance and many statistics educators (e.g., Rubin, Bruce, & Tenney, 1990;Shaughnessy, 1992; Tversky & Kahneman, 1971) have suggested that the samplingdistribution is a core idea in the understanding of statistical inference, somethingthat many teachers of the subject have intuitively recognised. In addition, manycurrent students of statistics are not mathematically trained, and hence the moreabstract concepts in statistics tend to be demonstrated rather than derived. Currentcomputer technology allows ideas such as the sampling distribution to bedemonstrated easily using readily available software. One need only look at theproliferation of computer activities dedicated to the Central Limit Theorem toconfirm this (e.g., Finzer & Erickson, 1998; Kader, 1990; Kreiger & Pinter-Lucke,1992; Stirling, 2002).The aim of this research was to produce empirical evidence to determine if theeducational emphasis on the sampling distribution holds the potential to enhancestudent understanding of statistical inference.Theoretical FrameworkThe study was concerned with the relationship between students’ knowledgeof the sampling distribution, and the level of understanding that theydemonstrated concerning statistical inference. In order to examine this it wasnecessary to consider what constituted knowledge in the content domain ofsampling distribution, and how this knowledge could be determined and
The Sampling Distribution and Understanding Statistical Inferenceevaluated. It was also necessary to propose a model for understanding, anddetermine how understanding of statistical inference would be measured.Procedural and Conceptual Understanding, and SchemasIt has been long recognised by many educators and researchers that oftenstudents are able to complete problems in statistics successfully but, at the sametime, demonstrate no real understanding of the concepts inherent in these tasks(Garfield, 2002; Garfield & Ahlgren, 1988; Williams, 1998). In investigating thissituation, particularly in relation to statistical inference, it is useful to differentiatebetween procedural and conceptual understanding. Procedural understandingdescribes a student’s ability to carry out routine tasks successfully, whereasconceptual understanding implies that the student understands what is being doneand why.In order to think about levels of student understanding it helps to adopt arepresentation for the structure of knowledge. In this study learning was viewedfrom the constructivist position, where students are not regarded as passivereceivers of information but rather as active constructors of highly personal mentalstructures called schema (see, e.g., Howard, 1983; Piaget, 1970). Marshall (1995)writes:A distinctive feature of a schema is that when one piece of information associatedwith it is retrieved from memory, other pieces of information connected to thesame schema are also activated and available for mental processing. (p. vii)It makes sense, then, to think of the schema as a connected network of concepts.This model fits well with the role schemas play in the construction of knowledge.Hiebert and Carpenter (1992) suggest a useful relationship between the form ofthe cognitive structure and the level of understanding that is evidenced by thestudent. They suggest:Conceptual knowledge is equated with connected networks A unit ofconceptual knowledge is not stored as an isolated piece of information; it isconceptual knowledge only if it is part of a network. On the other hand, we defineprocedural knowledge as a sequence of actions. The minimal connections neededto create internal representations of a procedure are connections betweensucceeding actions in the procedure. (p. 78)This relationship between level of understanding and the form and complexity of astudent’s schema gives a theoretical justification for evaluation of the student’sunderstanding based on an analysis of the relevant schema. In order to undertakethis analysis, however, it is necessary to propose a model of a schema that could beconsidered to represent conceptual understanding in the relevant content domain,in this case statistical inference, including the sub-domain of interest, the samplingdistribution.Analysis of the Content Domain using Concept MapsBecause it is impossible to observe an individual’s schema directly, hypothesesare required about the structure of schemas appropriate for particular statisticaltasks. It is necessary to look carefully at the content of the task in order to ascertainthe knowledge required to carry out that task successfully. To investigate theimportant concepts in statistical inference an analysis of the underlying knowledgeand the way in which each of the component ideas relates to the others is crucial.271
272LipsonThis enables identification of the desirable features of a schema that will supportboth procedural and conceptual understanding in statistical inference.A useful tool for doing this analysis is the concept map, a technique developedby Novak and Gowin (1984) and used for the purpose of content analysis by someeducators (e.g., Jonassen, Beissner, & Yacci, 1993; Starr & Krajcik, 1990). Conceptmaps constitute a method for externalising the knowledge structure in a particularcontent domain. They are two-dimensional diagrams in which relationshipsamong concepts are made explicit. When two or more concepts are linked togetherwith a label then this forms a proposition, the formation of which is taken to indicaterecognition of that aspect of the concept. According to Novak (1990), “the meaningof any concept for a person would be represented by all of the propositionallinkages the person could construct for that concept” (p. 29).Constructing a concept map requires one to identify important conceptsconcerned with the topic, rank these hierarchically, order them logically, andrecognise relationships where they occur. In this research the concept map wasused to analyse the content domain of statistical inference, making explicit theconcepts and relationships between concepts that are fundamental to developingunderstanding in this topic, in particular sampling distribution.In order to study and evaluate the students’ schemas an externalrepresentation of that mental structure was necessary. Once again, the conceptmap provides a method for obtaining external representations of an individual’sschema. Concept maps have been used in educational research to facilitate thestudy of the students’ schemas before and after instruction (Novak, 1990), and asan assessment tool to give insight into students’ understanding (Schau & Mattern,1997; Shavelson, 1993). By directing students to construct concept maps for variouscomponents of the course the researchers could gain insight into the relevantstudent schemas.Measuring Procedural and Conceptual UnderstandingSince this research was concerned with the development of procedural andconceptual understanding in introductory statistical inference, it was necessary todetermine a measure of conceptual understanding and a measure of proceduralunderstanding for each participant. Since procedural understanding is a commonfocus of many tasks assessing statistical understanding, there existed a variety oftasks that could be used to measure procedural understanding. For conceptualunderstanding, however, few tasks were available that had been trialed andvalidated, and that covered the content domain under investigation. For this studysuch instruments needed to be developed.From an analysis of what it means to know and understand mathematicsPutnam, Lampert, and Peterson (1990) proposed that there are five key themesunderpinning mathematical understanding. These are: Understanding asRepresentation, Understanding as Knowledge Structure, Understanding as Connectionsbetween Types of Knowledge, Understanding as the Active Construction of Knowledge,and Understanding as Situated Cognition. These themes were taken by Nitko andLane (1990) and related to the development and measurement of understanding instatistics. Their framework was further developed by the researcher (Lipson, 2000)to develop a range of tasks to assess aspects of either procedural or conceptualunderstanding in the particular knowledge domain of introductory statisticalinference, as shown in Table 1.
The Sampling Distribution and Understanding Statistical InferenceTable 1Framework for Developing Tasks to Measure Understanding of Statistical InferenceKey themeUnderstanding asRepresentationUnderstanding asKnowledge StructureUnderstanding asConnections between Typesof KnowledgeUnderstanding as the ActiveConstruction of KnowledgeUnderstanding as SituatedCognitionRelated assessment tasksTasks involve application of standard notation,representation, and algorithms to solve statisticalproblems. This would include standardapplications of the t-test or chi-square test forexample.Tasks give insight into the knowledge structures ofstudents; that is, tasks demonstrate that the studenthas made a connection between concepts, asdemonstrated by hypothesis testing and use ofconfidence intervals.Tasks require students to integrate formalknowledge with informal knowledge developedoutside the class. This would include tasksrequiring the interpretation of statistical concepts.Tasks enable the teacher to monitor thedevelopment of knowledge over time, such as thecreation of concept maps.Tasks require students to apply their knowledge ina variety of contexts, different from thosepreviously seen and discussed in the classroom.In this study, tasks that were developed within the classification ofUnderstanding as Representation were considered to measure proceduralunderstanding, as they refer to applications of standard procedures. Tasks thatwere developed under the other four key themes of understanding wereconsidered to contribute to the measurement of conceptual understanding, as theyrequired the students to have an holistic view of the processes that underliestatistical inference, their purpose, and relationships. Using this framework as aguide, known work on assessment at this time was expanded and supplementedby the researcher to give a set of tasks that covered the suggested range of aspectsof understanding over the full content domain.Research HypothesesThis study was concerned with the relationship between students’ knowledgeof the sampling distribution, evidenced by analysis of their concept maps, and thelevels of procedural and conceptual understanding that they demonstratedconcerning introductory statistical inference, measured by the tasks developedusing the framework in Table 1. The research hypotheses can thus be stated asfollows:1.Those students whose schema for sampling distribution demonstratedlinks to the sampling process and whose schema for statistical inferenceincluded links to the sampling distribution, would demonstrate the highestlevels of conceptual understanding of introductory statistical inference.273
274Lipson2.The level of procedural understanding demonstrated by students wouldnot necessarily be related to the content and form of their schemas forsampling distribution and statistical inference.MethodThe Setting of the StudyThe study was conducted at an Australian university. The 23 mature-agestudents were undertaking graduate studies in either Social or Health Statistics.They were generally graduates from other disciplines, such as Business or Nursing,who had determined that knowledge of statistics would be helpful for them intheir future careers. The study took place during the conduct of a subject called AnIntroduction to Statistics which was taught over a 13 week semester, and classeswere held one evening a week for 3.5 hours. All data for the study were collectedduring the final 6 weeks of the course, and in the examination held one week afterthe end of the course.Content Domain AnalysisIn order to provide a benchmark for the evaluation of the student conceptmaps, the researcher and a colleague—both content experts in the area ofintroductory statistical inference—constructed a series of concept maps, for thesampling distribution, hypothesis testing, estimation, and statistical inference.These maps were first constructed by each of the experts individually and then, bya process of negotiation, common maps were agreed upon. These were termed theexpert maps. From these expert maps, certain key propositions could be identified,which summarised both the knowledge domain and the connections betweenaspects of knowledge, and which characterised a connected schema. The expertconcept map for the sampling distribution is shown in Figure 1 and thepropositions identified from this are given in Table 2. Propositions are identifiedfrom the concept map by taking each pair of concepts together with the linkingwords and forming a statement.
The Sampling Distribution and Understanding Statistical InferencePopulationParameterseg p µ, !is describedby275PopulationgivesSamplefrom which wecan calculateSampleStatisticseg p, x, rwhich arewhich showSamplingVariabilityConstantwhich can besummarised SpreadindicatingShapeclose todepends onSampleSizeNormalDistributionFigure 1. Expert concept map for sampling distribution.Table 2Key Propositions Identified in the Expert Concept Map for Sampling DistributionPropositionsSamples are selected from populations.Populations (distributions) are described by parameters.Parameters are constant in value.Samples are described by statistics.Statistics are variable quantities.The distribution of a sample statistics is known as a sampling distribution.The sampling distribution of the sample statistic is approximately normal.The sampling distribution of the sample statistic is characterised by shape, centre, spread.The spread of the sampling distribution is related to the sample size.The sampling distribution is centred at the population parameter.
276LipsonAnalysis of Student Schema DevelopmentPrevious research (Jonasson, Beissner, & Yacci, 1993) has shown that, asstudents learn, the schema they create becomes closer in structure to those of theirinstructors, and thus that the students’ knowledge structure can be evaluated bycomparing the students’ concept maps with the expert maps. During this study thestudents were asked to prepare concept maps for the following topics,approximately one map each week until the final session of the course.1.2.3.4.5.6.The sampling distribution of the sample proportion.The sampling distribution of the sample mean.The sampling distribution (general).Hypothesis testing.Estimation.Statistical inference.In order to facilitate the mapping process, the students were provided with alist of the key terms that had been derived from the expert maps, and then thestudents were asked to use these terms in the constructions of their own maps.Students were advised that the terms listed were only suggested, and any could beomitted or others added as required. The list of terms used as a starting point foreach of these maps is given in Appendix 1. The purpose of the concept mappingexercises was to document the students’ schemas at particular points in time. Theauthor could identify from the maps the propositions formed by relating the termsgiven and then evaluate the student maps by comparison with the expert maps.This comparison was carried out not only in terms of both the number and type ofpropositions present, but also in terms of the links between various propositions.From a qualitative analysis of the propositions evidenced by the series of sixconcept maps prepared by the students over the period of the study students werecategorised into groups by the relative closeness of their association to the expertmaps described in the previous section and by change over time. Of importancewas not merely the number of propositions included, but which ones they were.More details about this process have been reported elsewhere (Lipson, 2002).Measures of Procedural and Conceptual UnderstandingSeveral tasks were used to measure procedural and conceptual understanding.Some were based on the Statistical Reasoning Assessment instrument developed toassess conceptual understanding in probability and statistics by Konold andGarfield (1993). This series of multiple choice questions built on earlier work ofKonold and others (Falk, 1993; Kahneman & Tversky, 1972; Konold, 1991). Thetasks used in the study were designed to measure procedural and conceptualunderstanding in statistical inference, over the relevant content domain. To ensureall content areas were covered, there were seven tasks pertaining to themeasurement of procedural understanding, each one concerned with a differentsub-section of the content domain. Example 1, shown in Figure 2, a routineproblem concerned with the measuring the students’ ability to carry out a standardt-test from first principles, is an example of such a task.Four tasks were used to measure conceptual understanding. Examples of threeof these tasks, classified according to the framework of Nitko and Lane (1990), aregiven in Examples 2, 3, and 4 in Figures 2 and 3. Adapted from the StatisticalReasoning Test (Konold & Garfield, 1993), based on earlier work of Kahneman and
The Sampling Distribution and Understanding Statistical InferenceTversky (1972), Example 2 was classified as Understanding as Connectionsbetween Types of Knowledge. Tversky and Kahneman (1972) found that 56% ofundergraduate students incorrectly gave the answer C, suggesting that themajority of students believe that the variability of the sampling distribution isindependent of the sample size. Obtaining the correct answer, B, implies that thestudent appreciates that the variability in the sampling distribution of the sampleproportion is larger when the sample size is smaller. Response A indicates that thevariability of the sampling distribution of the sample proportion is seen to increasewith the sample size.Example 1 (Procedural knowledge)According to a Census held in 1956, the mean number of residents per household in aninner suburb, Richthorn, was 3.6. In 1995, a student randomly sampled 11 households fromthe suburb and recorded the numbe
The Sampling Distribution and Understanding Statistical Inference 271 evaluated. It was also necessary to propose a model for understanding, and determine how understanding of statistical inference would be measured. Procedural and Conceptual Understanding, and Schemas It has been long recognised by many educators and researchers that often
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